1/3+1/6+1/10+1/5+.+1/5050

1/3+1/6+1/10+1/5+.+1/5050
1/3+1/6+1/10+1/5+.+1/5050

1/3+1/6+1/10+1/5+.+1/5050
观察分母,发现3=1+2,6=1+2+3,.,5050=1+2+.+100.
为等差数列的和,以3为第1项S1,即Sn=(n+2)(n+1)/2,一共有99项,
故1/Sn=2/(n+2)(n+1)=2[1/(n+1)-1/(n+2)],
原式=2*(1/2-1/3)+2*(1/3-1/4)+.+2*(1/100-1/101)=2*(1/2-1/101)
=1-2/101=99/101.

1/3 + 1/6 + 1/10 + 1/15 + ... ... + 1/5050
= 2 (1/6 + 1/12 + 1/20 + 1/30 + ... + 1/10100)
= 2 (1/2 - 1/3 + 1/3 - 1/4 + ... - 1/101)
= 2 (1/2 - 1/101)
= 99/101